"Eulerian" closed walk with of length less than v+e

by Miquel-point, May 16, 2025, 4:56 PM

Show that a connected graph $G=(V, E)$ has a closed walk of length at most $|V|+|E|-1$ passing through each edge of $G$ at least once.

Proposed by Radu Bumbăcea
combinatorics
graph theory
L

inequalities

by Ducksohappi, May 16, 2025, 12:57 PM

let a,b,c be non-negative numbers such that ab+bc+ca>0. Prove:
$ \sum_{cyc} \frac{b+c}{2a^2+bc}\ge \frac{6}{a+b+c}$
P/s: I have analysed:$ S_a=\frac{b^2+c^2+3bc-ab-ac}{(2b^2+ac)(2c^2+2ab)}$, similarly to $S_b, S_c$, by SOS
This post has been edited 1 time. Last edited by Ducksohappi, 5 hours ago
inequalities
L

A little problem

by TNKT, May 15, 2025, 1:17 PM

Problem. Let a,b,c be three positive real numbers with a+b+c=3. Prove that \color{blue}{\frac{1}{4a^{2}+9}+\frac{1}{4b^{2}+9}+\frac{1}{4c^{2}+9}\le \frac{3}{abc+12}.}
When does equality hold?
P/s: Could someone please convert it to latex help me? Thank you!
See also MSE: https://math.stackexchange.com/questions/5065499/prove-that-frac14a29-frac14b29-frac14c29-le-frac3
This post has been edited 1 time. Last edited by TNKT, Today at 1:23 AM
inequalities
L

Trigonometric Product

by Henryfamz, May 13, 2025, 4:52 PM

Compute $$\prod_{n=1}^{45}\sin(2n-1)$$
algebra
L

Gcd(m,n) and Lcm(m,n)&F.E.

by Jackson0423, May 13, 2025, 4:12 PM

Find all functions \( f : \mathbb{N} \to \mathbb{N} \) such that for all positive integers \( m, n \),
\[ f(mn) = \mathrm{lcm}(m, n) \cdot \gcd(f(m), f(n)), \]where \( \mathrm{lcm}(m, n) \) and \( \gcd(m, n) \) denote the least common multiple and the greatest common divisor of \( m \) and \( n \), respectively.
algebra
algebra unsolved
number theory
L

IMO Genre Predictions

by ohiorizzler1434, May 3, 2025, 6:51 AM

Everybody, with IMO upcoming, what are you predictions for the problem genres?


Personally I predict: predict
IMO
2025
IMO 2025
Predictions
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D1024 : Can you do that?

by Dattier, Apr 29, 2025, 5:11 PM

Let $x_{n+1}=x_n^2+1$ and $x_0=1$.

Can you calculate $\left(\sum\limits_{i=1}^{2^{2025}} x_i\right) \mod 10^{30}$?
This post has been edited 2 times. Last edited by Dattier, May 8, 2025, 8:14 PM
Computer solutions
number theory
L

f(x + f(y)) is equal to x + f(y) or f(f(x)) + y

by parmenides51, Jul 20, 2024, 11:49 PM

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying the following condition: for any real numbers $x$ and $y$, the number $f(x + f(y))$ is equal to $x + f(y)$ or $f(f(x)) + y$
functional equation
algebra
L

Two lengths are equal

by 62861, Jul 7, 2016, 11:09 PM

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$.

Proposed by El Salvador
This post has been edited 1 time. Last edited by 62861, Jul 20, 2016, 7:44 AM
Reason: added proposer
geometry
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The sequence does not contain numbers of the form 2^n - 1

by Amir Hossein, Sep 2, 2010, 1:04 AM

Prove that the sequence $5, 12, 19, 26, 33,\cdots $ contains no term of the form $2^n -1.$
number theory
power of 2
arithmetic sequence
IMO Shortlist
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The oldest, shortest words — "yes" and "no" — are those which require the most thought.

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